Is the Function f(x) = 2x/(x-1) Both Injective and Surjective?

In summary: i believe he's taking the discriminant of a quadratic.i believe he's taking the discriminant of a quadratic.
  • #1
mtayab1994
584
0

Homework Statement



f:]1,+∞[→]2,+∞[
x→ 2x/(x-1)


Homework Equations


Prove that f is injective and serjective.


The Attempt at a Solution



I already proved that it's injective by stating the injectivity law:
for every (a,b)ε]1,+∞[: f(a)=f(b) implies a=b

so: 2a/(a-1)=2b/(b-1) entails: 2ab-2b=2ab-2a entails -2b=-2a entails a=b

Can anyone please tell me how to prove that its serjective?
 
Physics news on Phys.org
  • #2
Sure. If y is in ]2,+∞[, calculate what x in ]1,+∞[ gives f(x) = y.
 
  • #3
mtayab1994 said:

Homework Statement



f:]1,+∞[→]2,+∞[
x→ 2x/(x-1)


Homework Equations


Prove that f is injective and serjective.


The Attempt at a Solution



I already proved that it's injective by stating the injectivity law:
for every (a,b)ε]1,+∞[: f(a)=f(b) implies a=b

so: 2a/(a-1)=2b/(b-1) entails: 2ab-2b=2ab-2a entails -2b=-2a entails a=b

Can anyone please tell me how to prove that its serjective?

In your own words, what does it mean to say that f is surjective? (That is sUrjective, not sErjective!) Turn that verbal statement into an equation and then work on the equation, to see what conclusions you can make, or else use some known, general properties to get a conclusion.

RGV
 
  • #4
i got it
f(x)=y
y=2x/x-1 equivalence y(x-1)=2x equivalence yx-2x-y=0
now we find Δ
Δ=4+4y^2
since Δ≥0 therefore there is some solution to this equation and therefore f is serjective.
 
  • #5
What does [itex] \Delta = 4 + 4y^2 [/itex] have to do with anything here? Anyway, you are still spelling surjective incorrectly.

RGV
 
  • #6
Ray Vickson said:
What does [itex] \Delta = 4 + 4y^2 [/itex] have to do with anything here? Anyway, you are still spelling surjective incorrectly.

RGV

Well my teacher stated that if we find that Δ≥0 then therefore f is surjective and btw my first language is english, but I'm learning overseas in Morocco and all the lessons here are in Arabic, so that's probably the reason why i spelled it wrong.
 
  • #7
Ray Vickson said:
What does [itex] \Delta = 4 + 4y^2 [/itex] have to do with anything here? Anyway, you are still spelling surjective incorrectly.

RGV

i believe he's taking the discriminant of a quadratic.
 
  • #8
Deveno said:
i believe he's taking the discriminant of a quadratic.

Of course he is; but where is the quadratic equation in this question?

RGV
 

FAQ: Is the Function f(x) = 2x/(x-1) Both Injective and Surjective?

What are polynomial functions?

Polynomial functions are mathematical expressions that involve variables raised to whole number powers and combined using addition, subtraction, multiplication, and non-negative exponents. They can be written in the form of f(x) = anxn + an-1xn-1 + ... + a1x + a0, where an to a0 are constants and n is a positive integer.

What is the degree of a polynomial function?

The degree of a polynomial function is the highest power of the variable in the expression. For example, in the function f(x) = 2x3 + 4x2 + 5x + 1, the degree is 3.

What is the leading coefficient of a polynomial function?

The leading coefficient of a polynomial function is the coefficient of the term with the highest degree. In the function f(x) = 2x3 + 4x2 + 5x + 1, the leading coefficient is 2.

What is the difference between a monomial and a polynomial?

A monomial is a single term with a coefficient and a variable raised to a power. A polynomial, on the other hand, is a sum of monomials. Monomials can be thought of as the building blocks of polynomials.

What are the different types of polynomial functions?

There are several types of polynomial functions, including linear (degree 1), quadratic (degree 2), cubic (degree 3), quartic (degree 4), and so on. There are also special types of polynomial functions such as constant (degree 0) and zero (degree 0 with no non-zero constants).

Back
Top